Since the BS of 6G wireless networks is equipped with a large-scale antenna array, massive device detection based on \(\mathbf {Y}\)  imposes prohibitive computational complexity. Considering \(\mathbf {X}\in \mathbb {C}^ {N \times M}\)  is simultaneously sparse and low-rank, namely its rank \(r^e = \text{rank}(\mathbf {X})\le \min \lbrace M,N\rbrace \) , we carry out dimension reduction before performing any sparse signal recovery. In particular, we can transform the original detection problem in (REF ) to a low-dimensional space, where the unknown matrix is of size \(N\times r^e\) . Consequently, the solution of the original problem can be recovered based on that of the low-dimensional problem. Specifically, we conduct dimension reduction as follows [1]}:
